The 4th term of a GP is 3/7.If the 7th term is eight times as much as the 4th term, find the third term of the progression.
In GP n-th term:
an = a • r ⁿ⁻¹
where
a = a1 = initial value
r = common ratio
In this case:
a4 = 3 / 7
a4 = a • r ³
a • r ³ = 3 / 7
a7 = a • r ⁶
a7 = 8 • a4 = 8 • 3 / 7 = 24 / 7
a7 / a4 = a • r ⁶ / a • r ³ = r ³
( 24 / 7 ) / ( 3 / 7 ) = 24 / 3 = 8 = r ³
r = ∛8
r = 2
Put this value in equation:
a • r ³ = 3 / 7
a • 2 ³ = 3 / 7
8 a = 3 / 7
a = ( 3 / 7 ) / 8 = 3 / 7 • 8
a = 3 / 56
a3 = a • r ² = 3 / 56 • 2² = 3 / 56 • 4 =
12 / 56 = 4 • 3 / 4 • 14 = 3 / 14
Let's assume the first term of the geometric progression (GP) is 'a', and the common ratio is 'r'.
Given that the 4th term is 3/7, we can write the equation:
a * r^3 = 3/7 (equation 1)
Also, given that the 7th term is eight times the 4th term, we can write the equation:
a * r^6 = 8 * (3/7) (equation 2)
To find the 3rd term, we need to find the value of 'r'.
Dividing equation 2 by equation 1 gives us:
(a * r^6) / (a * r^3) = (8 * (3/7)) / (3/7)
r^3 = 8
Taking the cube root of both sides, we get:
r = 2
Substituting the value of 'r' in equation 1, we can solve for the first term 'a':
a * (2^3) = 3/7
a * 8 = 3/7
Simplifying, we have:
a = (3/7) / 8
a = 3/56
The third term can be found by multiplying the first term by the square of the common ratio:
a * r^2 = (3/56) * (2^2)
a * r^2 = (3/56) * 4
a * r^2 = 3/14
Therefore, the third term of the geometric progression is 3/14.
To find the third term of the geometric progression (GP), we need to find the common ratio (r) first.
Let's assume that the first term of the GP is 'a', and the second term is 'ar'. Then the third term will be 'ar^2'.
We are given that the fourth term is 3/7, so we have:
ar^3 = 3/7 (equation 1)
We are also given that the seventh term is eight times the fourth term. The seventh term is 'ar^6', and since it is eight times the fourth term, we can write:
8 * (3/7) = ar^6 (equation 2)
To find the common ratio (r), divide equation 2 by equation 1:
(8 * (3/7)) / (3/7) = (ar^6) / (ar^3)
8 = r^3
We now know that the common ratio (r) is equal to 2. (Since 2^3 = 8)
To find the third term (ar^2), substitute the common ratio (r) into equation 1:
ar^3 = 3/7
Substituting r=2, we get:
a(2^3) = 3/7
a * 8 = 3/7
8a = 3/7
Solving for 'a', we can multiply both sides by 7/8:
a = (3/7) * (7/8)
a = 3/8
Finally, substitute 'a' and 'r' into the expression for the third term to find its value:
ar^2 = (3/8) * (2^2)
ar^2 = (3/8) * 4
ar^2 = 12/8
ar^2 = 3/2
Therefore, the third term of the progression is 3/2.